\chapter{Metrizable spaces}









\begin{theorem}
A metric space $(X,d)$ naturally induces a topological space by the basis $\mathcal{B}_d = \{ B(p,r) \subseteq X : p\in X \land r \in \mathbb{R}^{+} \}$.
\end{theorem}

So all metric spaces induce a topological space, however for which  topological spaces does there exist a metric space that generates said topology? This is the class of \emph{metrizable topological spaces}; topological spaces that can be reformulated as metric spaces.

\begin{proposition}
Let $(X,d)$ be a metric space. The open balls of $(X,d)$ are a basis for a topological space $(X,\mathcal{T})$. The resulting topological space is called the \emph{topological space induced by $(X,d)$}.
\end{proposition}

I may often refer (as an abuse of notation) to the metric space as its induced topological space. 

\begin{definition}[Metrizable topological space]
A \emph{metrizable topological space} is a topological space $(X,\mathcal{T})$ such that there exists a metric $d : X\times X \to [0,\infty)$ such thatthe topology induced by $d$ is $\mathcal{T}$.
\end{definition}


It is possible that such such a metric is not unique; Different metrics on the same set can possibly induce the same topology. They are said to be equivalent metrics if they both form the same topology.


Note that showing that countable unions of open balls of $d_1$ can be used to form any open ball of $d_2$ and vice versa means that each open ball in one metric is an open set in the other; this fact can then prove that they induce the same topology and hence are equivalent metrics.


\section{Metrizations}

Metric spaces are much nicer to work with than topological spaces, so given a topolgical space, mathematicians often want to see if it is metrizable.

We search for a set of minimal properties that a topological space can obey that can be used to construct a metric. Such results are called \emph{metrization theorems}; sufficient conditions for a topological space to be metrizable.

Note that this study makes use of the separation properties.


\begin{theorem}[Urysohn Metrization theorem]
\end{theorem}